Peg

Author

Eric W. Weisstein
January 15, 2007

This notebook downloaded from http://mathworld.wolfram.com/notebooks/RecreationalMath/Peg.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Peg.html.

©2007 Wolfram Research, Inc. except for portions noted otherwise

Graphic

Square[s_]:={{-1,-1},{1,-1},{1,1},{-1,1}}s
line[l_List]:=Line[Append[l,First[l]]]

Show[GraphicsArray[{
Graphics[{line[Square[1/2]],Disk[{0,0},1/2]},AspectRatio->Automatic],
Graphics[{Circle[{0,0},1/2],
Polygon[Square[Sqrt[2]/4]
]},AspectRatio->Automatic,PlotRange->All]
}]]

Plot

Rround[n_]:=Pi^(n/2)/Gamma[1+n/2]/2^n
Rsquare[n_]:=2^n Gamma[1+n/2]/n^(n/2)/Pi^(n/2)

Plot[Rround[n]/Rsquare[n],{n,0,10},AxesLabel->TraditionalForm/@{n,(Subscript[R, round])[n]/(Subscript[R, square])[n]},
PlotStyle->Red]

Rround[n]/Rsquare[n]

Solve[(((2^(-2 n)) (n^(n/2)) (π^n))/((Gamma[1+(n/2)])^2))==1,n]

n/.FindRoot[(((2^(-2 n)) (n^(n/2)) (π^n))/((Gamma[1+(n/2)])^2))==1,{n,8,8.2},WorkingPrecision->100]

V6.1

Reduce[(((2^(-2 n)) (n^(n/2)) (π^n))/((Gamma[1+(n/2)])^2))==1&&1<n<10,n]

FullSimplify[-(E^(-(1/2) (4 Log[2]-2 Log[π]-Log[#1]) #1))+((Gamma[1+((#1)/2)])^2)&[x]]

Root[{-(4^(-#)) (π^(#)) ((#)^(#/2))+((Gamma[1+((#)/2)])^2)&,8.1379410460913723765298389840532233724802091974718053826398`30.102999566398122}]//N